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# Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm - PowerPoint PPT Presentation

Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm. Lecture XXVI. Change in the Marginal Cost. Shares of Marginal Cost Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price.

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### Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

Lecture XXVI

Change in the Marginal Cost Cost and the Multi-Product Firm

• Shares of Marginal Cost

• Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price

Multiproduct Firm inputs as

• Expanding the production function to a multiproduct technology

• With lecture, allowing for multiple outputs.

• Multiplying through by differentiating the production function by a vector of output levels, holding prices and other outputs constant yieldsγyields

• Using the tired first-order conditions

• With differentiating the production function by a vector of output levels, holding prices and other outputs constant yields

• Collecting these equations: yields

• Differentiating the first-order conditions with respect to ln(z’)

• Differentiating the first-order conditions with respect to ln(p’)

• θ yieldsir is the share of the ith input in the marginal cost of the rth product.

• Summing this marginal cost over all inputs

• Expanding the differential model further, we introduce quasi-fixed variables into the production set